- 08-9 Hector E. Lomeli, Rafael Ramirez-Ros
- Separatrix splitting for 3D volume-preserving maps
Jan 15, 08
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Abstract. We construct a family of integrable volume-preserving maps in $\Rset^3$
with a bi-dimensional heteroclinic connection of spherical shape
between two fixed points of saddle-focus type.
In other contexts,
such structures are called Hill's spherical vortices or spheromaks.
We study the splitting of the separatrix under volume-preserving
perturbations using a discrete version of the Melnikov method.
Firstly, we establish several properties under general perturbations.
we bound the topological complexity of the primary heteroclinic set
in terms of the degree of some polynomial perturbations.
We also give a sufficient condition for the splitting of the separatrix
under some entire perturbations.
A broad range of polynomial perturbations verify this sufficient condition.
we describe the shape and bifurcations of the primary heteroclinic set
for a specific perturbation.